Mister Exam
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Identical expressions
- Cos((x/ two)-(pi/ twelve))= zero
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-
Similar expressions
- Cos((x/2)+(pi/12))=0
Cos((x/2)-(pi/12))=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
/x pi\ cos|- - --| = 0 \2 12/
$$\cos{\left(\frac{x}{2} - \frac{\pi}{12} \right)} = 0$$
Detail solution
Given the equation
$$\cos{\left(\frac{x}{2} - \frac{\pi}{12} \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\cos{\left(\frac{x}{2} - \frac{\pi}{12} \right)} = 0$$
This equation is transformed to
$$\frac{x}{2} + \frac{5 \pi}{12} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$\frac{x}{2} + \frac{5 \pi}{12} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$\frac{x}{2} + \frac{5 \pi}{12} = 2 \pi n$$
$$\frac{x}{2} + \frac{5 \pi}{12} = 2 \pi n + \pi$$
, where n - is a integer
Move
$$\frac{5 \pi}{12}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{2} = 2 \pi n - \frac{5 \pi}{12}$$
$$\frac{x}{2} = 2 \pi n + \frac{7 \pi}{12}$$
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 4 \pi n - \frac{5 \pi}{6}$$
$$x_{2} = 4 \pi n + \frac{7 \pi}{6}$$
$$\cos{\left(\frac{x}{2} - \frac{\pi}{12} \right)} = 0$$
- this is the simplest trigonometric equation
with the change of sign in 0
We get:
$$\cos{\left(\frac{x}{2} - \frac{\pi}{12} \right)} = 0$$
This equation is transformed to
$$\frac{x}{2} + \frac{5 \pi}{12} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$\frac{x}{2} + \frac{5 \pi}{12} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$\frac{x}{2} + \frac{5 \pi}{12} = 2 \pi n$$
$$\frac{x}{2} + \frac{5 \pi}{12} = 2 \pi n + \pi$$
, where n - is a integer
Move
$$\frac{5 \pi}{12}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{2} = 2 \pi n - \frac{5 \pi}{12}$$
$$\frac{x}{2} = 2 \pi n + \frac{7 \pi}{12}$$
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 4 \pi n - \frac{5 \pi}{6}$$
$$x_{2} = 4 \pi n + \frac{7 \pi}{6}$$
Sum and product of roots
[src]
sum
5*pi 7*pi - ---- + ---- 6 6
$$- \frac{5 \pi}{6} + \frac{7 \pi}{6}$$
=
pi -- 3
$$\frac{\pi}{3}$$
product
-5*pi 7*pi -----*---- 6 6
$$- \frac{5 \pi}{6} \frac{7 \pi}{6}$$
=
2 -35*pi ------- 36
$$- \frac{35 \pi^{2}}{36}$$
-35*pi^2/36
Rapid solution
[src]
-5*pi
x1 = -----
6
$$x_{1} = - \frac{5 \pi}{6}$$
7*pi
x2 = ----
6
$$x_{2} = \frac{7 \pi}{6}$$
x2 = 7*pi/6
Numerical answer
[src]
x1 = 60.2138591938044
x2 = -272.794962086714
x3 = -78.0162175641465
x4 = -8.90117918517108
x5 = -96.8657734856853
x6 = 91.6297857297023
x7 = 53.9306738866248
x8 = -71.733032256967
x9 = 16.2315620435473
x10 = -40.317105721069
x11 = -21.4675497995303
x12 = 41.3643032722656
x13 = -222.529479629277
x14 = -15.1843644923507
x15 = 72.7802298081635
x16 = 7285.87696245033
x17 = 22.5147473507269
x18 = 79.0634151153431
x19 = -52.8834763354282
x20 = -46.6002910282486
x21 = -34.0339204138894
x22 = -59.1666616426078
x23 = 3.66519142918809
x24 = -27.7507351067098
x25 = 35.081117965086
x26 = -65.4498469497874
x27 = 66.497044500984
x28 = 97.9129710368819
x29 = -90.5825881785057
x30 = 85.3466004225227
x31 = 9.94837673636768
x32 = 28.7979326579064
x33 = 160.744824108678
x34 = -84.2994028713261
x35 = -140.848070635942
x36 = -2.61799387799149
x37 = 47.6474885794452
x37 = 47.6474885794452